3. Elliptic PDE

The high-dimensional version of a TPBVP is an elliptic PDE. These typically represent a system in steady state. The most famous is Poisson’s equation, which in 2D is

\[ \Delta u = \partial_{xx}u + \partial_{yy}u = f(x,y), \quad (x,y) \in \Omega \subset \mathbb{R}^2. \]

This equation represents the steady state of a diffusive process. The function \(f\) might be called the loading function in some contexts. If \(f(x,y)\equiv 0\), we get Laplace’s equation.

For Poisson’s equation we need a boundary condition imposed on the entire boundary \(\partial \Omega\). It may be Dirichlet, which prescribes the value of \(u\), Neumann, which prescribes \(\pp{u}{n}\), or a mixture, and the type can change along the boundary.